Mathematics for the interested outsider

A Continued Rant on Electromagnetism Texts and the Pedagogy of Science

Here’s the fundamental reason for your discomfort: as a mathematician, you don’t realize that scalar and vector potentials have *no physical significance* (or for that matter, do you understand the distinction between objects of physical significance and things that are merely convenient mathematical devices?).

It really doesn’t matter how scalar and vector potentials are defined, found, or justified, so long as they make it convenient for you to work with electric and magnetic fields, which *are* physical (after all, if potentials were physical, gauge freedom would make no sense).

On rare occasions (e.g. Aharonov-Bohm effect), there’s the illusion that (vector) potential has actual physical significance, but when you realize it’s only the *differences* in the potential, it ought to become obvious that, once again, potentials are just mathematically convenient devices to do what you can do with fields alone.

P.S. We physicists are very happy with merely achieving self-consistency, thankyouverymuch. Experiments will provide the remaining justification.

The thing is, none of that changes the fact that you’re flat-out lying to students when you say that the vanishing divergence of the magnetic field, on its own, implies the existence of a vector potential.

I think the commenter is confusing my complaint with a different, more common one: the fact that potentials are not uniquely defined as functions. But I actually don’t have a problem with that, since the same is true of any antiderivative. After all, what is an antiderivative but a potential function in a one-dimensional space? In fact, the concepts of torsors and gauge symmetries are intimately connected with this indefiniteness.

No, my complaint is that physicists are sloppy in their teaching, which they sweep under the carpet of agreement with certain experiments. It’s trivial to cook up magnetic fields in non-simply-connected spaces which satisfy Maxwell’s equations and yet have no globally-defined potential at all. It’s not just that a potential is only defined up to an additive constant; it’s that when you go around certain loops the value of the potential must have changed, and so at no point can the function take any “self-consistent” value.

In being so sloppy, physicists commit the sin of making unstated assumptions, and in doing so in front of kids who are too naïve to know better. A professor may know that this is only true in spaces without holes, but his students probably don’t, and they won’t until they rely on the assumption in a case where it doesn’t hold. That’s really all I’m saying: state your assumptions; unstated assumptions are anathema to science.

As for the physical significance of potentials, I won’t even bother delving into the fact that explaining Aharonov-Bohm with fields alone entails chucking locality right out the window. Rest assured that once you move on from classical electromagnetism to quantum electrodynamics and other quantum field theories, the potential is clearly physically significant.

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Please don’t take him too seriously. I can’t imagine how any capable physicist could be so condescending and oblivious at the same time.

“After all, if potentials were physical, gauge freedom would make no sense.”

Yeah, sure.

Why not quote Feynman on the matter? (from the Feynman Lectures)

“Nevertheless, the vector potential A (together with the scalar potential ϕ that goes with it) appear to give the most direct description of the physics. This becomes more and more apparent the more deeply we go into the quantum theory. In the general theory of quantum electrodynamics, one takes the vector and scalar potentials as the fundamental quantities in a set of equations that replace the Maxwell equations: E and B are slowly disapearing from the modern expression of physical laws; they are being replaced by A and ϕ.”

You are basically right about both physics and mathematics. The problem, at least for most of teachers, is jamming all these information into very short period of time. When I started teaching I tried to be very careful to cover all loop-holes and assumptions, both physics and math. Students however cannot process that much of information in a given amount of time, and worse yet, introducing additional subtlety often confuse them more. For specific issue you quote in your blog, I usually briefly mention that the statement doesn’t hold for non-simply connected space then move on. There is simply no way to cover a non-trivial case detail enough not to confuse students.

Including all the careful loop holes and assumptions is very time consuming, but, in the example being discussed, we don’t need to get into simply connectedness and cohomolgy. Instead of saying “Since div B=0, and since B=curl A implies div B=0, we conclude there is an A such that B=curl A”, we can say “Since div B=0, and since div B=0 implies there is a vector A such that B=curl A, we conclude there is an A such that B=curl A”. I’ve taught several variable calculus several times, and this material has always been part of the syllabus.

On the subject of electromagnetism books lying about the underlying mathematics, as an undergraduate, I enjoyed learning from Griffiths’ text. He lies, but at least he includes helpful footnotes saying “Actually, this argument is completely illogical. The result is true, and a correct proof exists”.

Delurking to join the voices supporting John, and to echo Dan Piponi’s comment. (Given that the commenter seems to have completely missed the point of John’s post, it’s not sure what else can be said.)

The sloppy mathematics that is employed in physics pedagogy is the primary reason that, as an undergraduate, I changed my major from physics to math. At the time, the mathematics in the physics texts that I encountered made no sense; as I learned more mathematics I understood why the physics text were inscrutable to me: because the math therein was presented with so little rigor that OF COURSE it wouldn’t make sense with all of the significant details elided. That being said, the real mathematics behind physics is very interesting and it is a shame that SO MANY so-called mathematical physics texts do such a HORRIBLE job of explaining the math. One would hope for lots of concrete examples and calculations but, more often than not, these texts are simply a compendium of theorems and definitions with “intuitive” explanations (if you’re lucky) for why a given theorem is true.

You need to understand that we physicists treat our scientific models, as do any other scientists for their respective models, as approximate reflections of natural phenomena, not actual realizations of it. With that in mind, we throw some mathematical formalities in the back of the closet, at least those technical assumptions which cannot be verified empirically. That means that the assumption that the electrostatic field is irrotational is enough to assume that it is also conservative. This is because it is impossible to determine experimentally whether or not the electrostatic field produced by some system of stationary charges is defined on a simply-connected space. Therefore, such an inquiry does not make sense in a scientific context.

Let me give a simpler example. Perhaps you have read in a text on mechanics that “force produces acceleration, which is the second time-derivative of a particle’s positional function.” You may think: “How DARE they give this law without first axiomatizing that the affine positional function of a particle is at least twice differentiable?!” Of course, that sort of complaint is made without realizing that it is impossible to empirically determine, to any order, the differentiability of a function in a scientific model. Therefore, we suppose that all our mappings, manfolds, etc., are analytic if such a notion exists. There is no reason not to do so.

This is the same reason why physicists used the Dirac-delta function in spite of decades of protests from mathematicians. We don’t care about the technicalities because they are unimportant.

It is surprising to me that several (apparent) physicists are coming out in defense of mathematical sloppiness, let alone in such a condescending way.

For hundreds of years physics has been built using the language of mathematics. While one might ask philosophically whether this is strictly necessary, in history and practice there is no question: modern physics makes thorough use of various mathematical structures and theorems, from the elementary to the not-so-elementary level. Should physicists apply mathematical formalisms correctly? Should they have some understanding of the mathematics they are applying? I find it ridiculous that some are arguing “Not necessarily” and “Well, if we make a mistake, experiments will let us know.”

In any walk of life we could make sloppy arguments with the thought that if we are wrong in an essential way, the consequences will become apparent to us sooner or later. This is probably true, but guess what’s even better — learning how to make correct arguments in the first place!

I find these arguments especially discouraging because we are not talking about (so to speak!) rocket science here, but rather multivariable calculus. When I taught engineers multivariable calculus I showed them the proof that irrotational vector fields in R^3 are conservative and mentioned that the simple connectedness of the domain is used in the proof and must be since otherwise there are counterexamples. (Maybe these counterexamples are “physically relevant” and maybe they’re not, but the idea that one can be a priori convinced that there can be no physically relevant counterexamples seems fishy to me.) My point is this: the (standard, not overly theoretical or ambitious) multivariable calculus textbook I used talked a bit about simple connectedness and the stronger students in my class were able to understand and take in the relevance of this hypothesis. Is someone who does not have enough patience or mathematical skill to succeed in a sophomore level multivariable calculus class really going to make a good physicist? Are the subtleties of the physical universe really less tricky or abstruse than these rudimentary topological concepts? I don’t think so. A good physicist is going to be good at undergraduate level mathematics. A physicist who argues in favor of extreme mathematical sloppiness does not inspire confidence.

It doesn’t really surprise me, Pete. The most vociferous math-haters I knew in any school I attended as a student were the engineering, computer science, and physics students.

That said, most of them — like Mike, most recently — do have a point in that mathematical rigor is secondary in physics to making accurate predictions; if treating the Dirac delta distribution as a “function” helps speed calculations that lead to predictions that agree closely with experiment, great.

The problem is that’s not what I’m calling out. I’m calling out a trend in pedagogy — in teaching something incorrect without even mentioning that you’re being sloppy for the sake of clarity. I’m not even saying that the details should be rigorously established in such a class; I only ask that the fact be mentioned.

Your position seems more generous than mine. (And the idea that a physics student would hate math is pretty baffling to me, although admittedly the only physicists I know are very close to being mathematicians. I’m guessing it’s more of a sociological phenomenon than an intellectual one: i.e., if you’re doing something that’s close to math but not quite, than that off-math thing is certainly “cooler” than math — since you’re doing it — and you want to promote it as the coolest thing.)

But I see no more merit in being sloppy in one’s work than being sloppy in one’s teaching. I understand that physics as a subject is not math and is not subsidiary to math: i.e., when you’re on the cutting edge you can’t wait around for mathematical rigor, because that could take decades. But the lack of mathematical rigor is something that you have to accept in order to do your job as a physicist. It’s not what makes physics cool — is it!?! — it’s part of what makes physics hard. But when you can have mathematical rigor, what is gained by turning your back on it? Isn’t it a silly affectation to turn your back on sophomore level mathematics?

My picture of a physicist is someone who is fiendishly clever in a way that mathematicians somehow do not have to be. Things move much more quickly in the physics world than in the mathematics world: a lot of people are working in semi-collaboration, semi-competition on a relatively small number of highly important problems. You need to think quickly and accurately in order to keep up with the pack, let alone make your mark. In this high pressure environment, are we really putting our money on people who have less than a B+ understanding of multivariable calculus? Of course not: surely the best physics people are better at math than the average physics people, because, well, smart people tend to be smart.

(I guess the real answer is that the best physicists are smart enough so that they can afford to look like they are being sloppy, while the light of their inner genius saves them from making some true mathematical gaffe. But then — as you say! — there remains a pedagogical problem: being taught by brilliant but sloppy-looking people who scoff at mathematical niceties BECAUSE they know how not to get tripped up by them do not necessarily make good teachers.)

Let me just clarify that when I talk about the off-math people needing to think that their specific brand of off-math is the coolest, I don’t mean to be condescending. Most of us think that, although many things are cool, whatever it is that we do is absolutely the coolest. That’s totally subjective and self-serving and also natural and healthy. The person who doesn’t think that what they do is the coolest is the one who seems weird to me.

The attitude in all three fields seems to be something along the lines of, “math is the annoying low-level language” which serves their own particular endeavors.

Engineers can point to real, physical things they’ve built; calculus may be essential to the design of a suspension bridge, but nobody is doing integrals on the construction site, especially since the advent of numerical computation systems.

Physicists can point to the fantastic realms of astronomy, or to the easily-popularized — and, yes, mangled — predictions of quantum theory or relativity. There’s a cottage industry devoted to “explaining” these subjects without showing a single equation.

Computer scientists deal with areas of mathematics so unlike those along the common track that it’s easy for some to forget that they’re actually math in the first place. To the extent that they do need “standard” mathematics, it’s increasingly bound up in well-established libraries that are so thoroughly designed for use as building blocks that almost nobody really needs to consider the tradeoffs of such implementations again beyond choosing which particular block to use.

What they all have in common is a sense that mathematics is the overly-strict schoolmarm who insists they eat their vegetables before going outside to play, with no sense that the vegetables might actually be good for them.

I should clarify that I’m speaking of these attitudes mostly among students; actual practitioners are less likely to run into mathematicians in the first place, less likely to express such “anti-social” viewpoints unprompted, and might even have realized that mathematics is actually worthwhile.

I agree with your sentiment. Though, this is a sign of a much bigger problem. Many fields have a more “eyes on the prize” approach. The concentration seems to be more on concepts that are new in the field of study, and the logical basis of which to approach them is treated more of a hindrance than a help.

A common joke in college math courses is: “Cool we’ll be studying math developed in this past century, just a few more classes and we’ll learn something developed 50 years ago.” The syntax is terrible but my overall point is that there are many times other fields forget that one must traverse through the muck to appreciate and understand the newer, more “shiny” concepts.

This is true; almost all of the mathematics involved in any undergraduate physics course — even ones that get into abstract algebra and algebraic topology — is a hundred years old or more. And most of the stuff that’s “only” a hundred years old (and the little that’s more recent in more advanced physics) is basically just different, more efficient ways of framing the older stuff.

But why is that such a bad thing to physics students (and teaching departments!)? Some of the most popular current techniques in computer science are “just” organizational tools. There’s really little in Scala that wasn’t in FORTRAN ’77; indeed, since both are Turing-complete languages neither can say more or less than the other. But Scala’s object-oriented and functional design makes organizing the same code much easier than it used to be. Just as GOTO is now widely “considered harmful”, so is explicit index-juggling, and more physics texts are getting on board with that sort of idea in their teaching.

And yet the problem with sweeping mathematical caveats under the rug is different. It’s not that there are two mathematical ways of describing the physics, and one is more current and elegant. The stated approach sands off the details of the mathematical model that make it correct but, yes, in many situations don’t actually affect the phenomenology.

So if describing “cool” phenomena like black holes and entanglement are what you really care about — and they seem to be all most physics undergrads actually care about — screw the details of the model. You get some great cocktail-party discussions, but you’ve stopped actually doing science.

I was troubled by this phenomena as an undergraduate and a graduate student. Having never formally studied differential geometry, I never encountered a proper explanation until I read the following paper:

I donot know — whether there is a way or not, to work in classical electrodynamics free from meaningless Dirac delta function. If yes, then some texts should be written as such, and the book should be popularized. Many parts of modern physics, and chemistry are at its infancy stage. These topics are not well-formalized. This is one reason of ugly mathemamatics occuring there. Unfortunatly, modern mathematicians are more culprit to this ugly situation that others. Mathematicians are in the habit of writing everything in a CONCISE way, providing proofs full of gaps, without considering the reader’s problem in understanding the content. Only proper writing with ALL details (not the faulty easy writing) in mathematical literature will help other sciences.

What you say about the concise writing style prevalent in mathematics is certainly true, and in my study I have often encountered the situation you mentioned: the author asserted, but did not prove, something which was not obvious to me.

However, the issue of how much detail to include in one’s expositions is a very delicate one, and the decision depends very much on the target audience. If I were writing a textbook for a first course in analysis, I should expect the reader to be fairly new to rigorous proofs, and would spell out the proofs in complete detail. On the other hand, if I were writing an advanced functional analysis textbook, I would be content to mention terms like “3-epsilon argument” or “diagonal argument” without providing details.

I should also point out that providing excruciating detail is harmful in at least one respect: that it can obscure the main ideas of the proof, and makes it hard to separate between key decisive steps and routine trivial arguments. This in turn slows down understanding, and can make the exposition painfully boring to read

If you encounter a missing step in a proof, why not treat it as an exercise to fill in the gap? Doing so may help deepen your understanding of the subject (remember than mathematics is learnt by doing, not watching). And if you are still stuck, consult an expert or ask around in forums.

I think you have missed the point I wanted to emphasis. There is a way to write everything without boring the reader. If you are really interested in seeing such writings, you should read two thin books: “Solutions to Weatherburn’s Elementary Vector Analysis–Rajnikant Sinha”, or “Basic Set Theory” by the same author. It is the foremost duty of the book author that every genuine reader should be able to understand every sentence of the book without any outside assistance. If one does not know the way, or if one has not so much time (the later being more likely), he should abandon the idea of writing a math book. There are people, although less in number, who are masters in this job. Let them do it. The reasons endorsed above against the compliability of this idea, are filmsy, not genuine, and only well-known excuses of maths authors. If every detail of certain proof of a theorem is presented, then mathematics would become most easy, most interesting, and most demanding subject in the world. For this, authors should have well-intention, and willing to do labor. There is nothing like “excruciating details”. Math book authors should presume that every reader of this book is “more intelligent” than him. After understanding some so-called difficult theorem, all of us have come across the feel that this theorem do not contain anything that “I can not understand”. Everytime if a motivated reader gets stuck, it is the fault of author, and only author. Math forums should have different purposes.

A Physicist and Mathematician sitting next to each other on a plane low flying over Wisconsin . The Physicist, in the window seat, alerts his fellow traveller to something of interest seen through the window.

Physicist: “Look, there are black cows in Wisconsin”

Mathematician:”No, when viewed through this window, while flying over this part of Wisconsin at this time of day, the superior surface of a cow shaped object is perceived as black by you”

….

This could irritate both or neither…just thought it fit the ‘mood’ of the comments…
Obviously no physicist would ascribe color as an attribute of an object and no mathematician declares more assumptions than are sufficient for an argument….

Your criticism is completely justified, but sadly, this practice has pervaded all of Physics. I believe that this is not just a pedagogical error, but an error in understanding and consequently, an error in exposition. Moreover, I would even go on to opine that the number of smart Physics people is very small!

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